Generalized Solutions of Parrondo's Games

Abstract In game theory, Parrondo's paradox describes the possibility of achieving winning outcomes by alternating between losing strategies. The framework had been conceptualized from a physical phenomenon termed flashing Brownian ratchets, but has since been useful in understanding a broad range of phenomena in the physical and life sciences, including the behavior of ecological systems and evolutionary trends. A minimal representation of the paradox is that of a pair of games played in random order; unfortunately, closed‐form solutions general in all parameters remain elusive. Here, we present explicit solutions for capital statistics and outcome conditions for a generalized game pair. The methodology is general and can be applied to the development of analytical methods across ratchet‐type models, and of Parrondo's paradox in general, which have wide‐ranging applications across physical and biological systems.

Suppose that out of n rounds, n + result in wins, n 0 result in draws, and n − result in losses. At steady-state, the average outcome probabilities ares = ω 1 s 1 + (1 − ω 1 )s 2 for s ∈ {p, q, r}, where ω 1 is the stationary distribution of capital state S 1 . The distribution P n (k) representing the probability of having k ∈ Z ∩ [−n, n] capital on round n can thus be written where the summation occurs over the solution set of simultaneous Diophantine equations n + +n 0 +n − = n and n + − n − = k. Such a solution set can be parametrized as (n + , n 0 , n − ) = (u + (|k| + k)/2, n − |k| − 2u, u + (|k| − k)/2 where u ∈ Z and 0 ≤ u ≤ (n − k)/2 ). The summation in Eq. (A4) is thus over u, enabling the closed-form calculation of P n (k). The expected capital µ(n) can be computed from this explicit capital distribution as which is identical to the result obtained in Eq. (5) of the main paper.

A4. Fundamental Matrix
We have Z = (I − H + Ω) −1 , but to simplify calculations, the identity Z(I − H) = I − Ω is used. This produces the set of equations and Z i3 as boundary conditions. Invoking Eqs. (A6a) and (A6b) eliminates Z i2 and Z i3 , and the normalization constraint We have Z ij = Z i1 for j = 1, and for 2 ≤ j ≤ M , the general solution is where R(x) is the unit ramp function.
Appendix B: Three-state M -branch game pair

B1. Conditions
Every M consecutive states is termed a tier. Winning a tier necessitates winning across all M branches; furthermore, an arbitrary number of losses l i at each state S i is allowed, so long as there is a corresponding number of wins to compensate. The probability of winning and losing a tier, respectively p and q, is thus where s i = γs+(1−γ)s i for s ∈ {p, r, q} and i ∈ [1, M ] are the mixed transition probabilities. The game is winning, fair, and losing when p > q, p = q, and p < q respectively. Cancellation of terms yield the simplistic condition in Eq. (8) of the main paper.

B2. Stationary Distribution
The eigenvector equation ω = ωH produces the set of equations where m ∈ [2, M −1]. But, as the recurrence in Eq. (B2b) involves non-constant coefficients, the usual method of solving the characteristic polynomial cannot be used. Instead, a tracking method can be used on the recursion tree to arrive at where F and G are counting functions as written in the main paper. Invoking Eq. (B2a) to eliminate ω 2 and the normalization constraint M m=1 ω m = 1 to set ω 1 then yields the solution for ω m as presented in Eq. (10) of the main paper.

B3. Fundamental Matrix
Again, the identity Z(I − H) = I − Ω is used. This produces the set of equations where i ∈ [1, M ] and j ∈ [2, M − 1]. As the recurrence in Eq. (B4b) is non-constant, the method of characteristic polynomials cannot be applied. The recurrence tree is tracked to give where Invoking Eq. (B4a) to eliminate Z i2 and the normalization constraint M j=1 Z ij = 1 to set Z i1 then yields the solution Z ij = F [1] j + ΛG [1] j + δ 1j Z * i − i G [1] j + T